Logical tensors and physical layouts

For a two-dimensional strided buffer, multiply each logical index by its element stride, add the base offset, then multiply by bytes per element for a byte displacement. Bounds and storage extent still matter. Strides may describe a dense array, a slice, a broadcasted view, or another mapping with aliasing. A shape alone does not establish contiguity, uniqueness, alignment, or how many allocated bytes the view owns. Distinguish element strides from byte strides in every formula.

A transpose view can exchange shape dimensions and strides while retaining the same underlying bytes. That is different from copying elements into a new dense buffer with transposed logical order. Whether a consumer accepts the view depends on its layout contract and the compiler's ability to lower the access pattern. A library call may require or prefer a specific arrangement, making a conversion worthwhile or necessary even when a mathematical operation itself is layout independent.

An elementwise consumer can often follow a producer's chosen layout, avoiding a conversion. A reduction, matrix product, or external library boundary can impose different access or layout requirements. If one value has several consumers, a layout favorable to one may disadvantage another. Propagation must account for the whole use graph, including copies, packing, and lifetime extension. The cheapest representation for one operation in isolation can increase total execution time or memory.

Layouts also affect which lanes access adjacent memory and how tiles are distributed across hardware. A physical address permutation that is legal for values may change coalescing, bank behavior, or vector loads. These are target-specific performance questions, not logical tensor equivalences. OpenXLA documents shape and layout separately; exact internal attributes and pass behavior can evolve. Record the compiler revision and target when measuring layout decisions instead of treating a current dump format as a permanent API.

Test non-square shapes, sliced inputs, nonzero offsets, and consumers with explicit layout restrictions. A square matrix or a corner index can accidentally yield the same offset under different mappings, hiding a mistake. Compare every logical output with a reference that understands the same view semantics. Inspect allocation and copy behavior separately from value correctness: removing a conversion is a performance result only if every consumer still receives the required values under valid storage and lifetime rules.

The matrix lab offers only a geometric contrast. Reset and set both a=1, b=1, predict determinant 1 and the shear, then Step. Its continuous two-dimensional transform does not describe a tensor's strides, lane distribution, or allocated byte order. Playback interpolates geometry while readouts retain the final selected matrix. Explain why preserving area in that diagram cannot prove that an arbitrary tensor layout preserves logical indexing or satisfies a downstream library's storage contract.